The time horizon \(T_\text{est}\)  corresponds to the estimation horizon, i.e. the number of steps in the history to estimate. In practice, the horizon is shifted forward every step when the number of observed states exceeds the time horizon \(T_\text{est}\)  to maintain computational tractability, particularly in realtime applications. This approach is why MHE is often referred to as the dual problem to MPC [1]}, as the same receding horizon approach is adopted to solve realtime tasks using solvers such as DDP [2]}. In the MPC problem, the goal is to find the sequence of controls that minimizes a finite-horizon cost function of the form
\(\mathcal {J}_{\text{mpc}} (\mathbf {U}) & = \sum _{t = 1}^{T_{\text{mpc}}} \mathcal {L}(\mathbf {x}_t, \mathbf {u}_t) + \phi (\mathbf {x}_{T_{\text{mpc}} + 1}).\) 
